Particles Emitting only a "Circle"
Your particles will emit in the direction of the face or the tangent (outward pointing ray direction relative to the surrounding vertecies) of the vertex they’re coming from depending if which emission method you’ve chosen (faces or vertex, volume is only necessary if you wand particles generated inside an object). All that is to say, if you have a sphere, you’re particles will emit in the shape of a sphere. A cube, cubed shaped. The random amount does what you think, adds randomness to the direction each particle emits from.

To get the arc you want, your emitter object should be an arc. Use a cylinder and delete the top and bottom flat face, and the degree of the arc will depend on how much of the curved cylinder wall you keep. Delete one half of it, you’ll have a 180 degree arc. Keep the whole cylinder 360 degrees and your arc will bee 360 degrees, etc.